Accelerometers on Quadrotors : What do they Really Measure?

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Accelerometers on Quadrotors : What do they Really Measure?

P Martin

To cite this version:

P Martin. Accelerometers on Quadrotors : What do they Really Measure?. Aerospace Lab, Alain Appriou, 2014, p. 1-10. �10.12762/2014.AL08-06�. �hal-01184755�

Aerial Robotics

P. Martin

(Mines ParisTech)

E-mail : philippe.martin@mines-paristech.fr DOI : 10.12762/2014.AL08-06

Accelerometers on Quadrotors : What do they Really Measure?

A revisited quadrotor model is proposed, including the so-called rotor drag. It differs from the model usually considered, even at first order, and much better explains the role of accelerometer feedback in control algorithms. The theoretical derivation is supported by experimental data.

Introduction

Quadrotor control has been an active area of investigation for several years. On the one hand, the quadrotor has several qualities, among them its very simple mechanical design, and qualifies as a viable con- cept of mini Unmanned Aerial Vehicle (UAV) for real-life missions ; on the other hand, it is perceived in the control community as a very rich case study in theoretical and applied control. The first control objec- tive is to ensure a stable flight at moderate velocities and, in particular, in hovering; this fundamental building block is then used to develop higher-level tasks.

However, for experiments designed to work only in the lab with an off-board measuring device, e.g. [1], quadrotors all rely at the heart on strapdown MEMS inertial sensors (gyroscopes and accelerom- eters). These inertial sensors may be used alone (as far as horizontal stabilization is concerned) [2], or supplemented by other sensors, which usually provide some position-related information. Representa- tive designs are: ultrasonic rangers [3]; (simple) GPS module when outdoors and infrared rangers when indoors [4] ; carrier phase differ- ential GPS [5]; laser rangefinder [6]; vision system [7], [8], [9] ; laser rangefinder and vision system [10], [11]. Unfortunately those extra sensors have inherent drawbacks (low bandwidth, possible tempo- rary unavailability, etc.), hence inertial sensors remain essential for basic stabilization.

Nearly all of the papers in the literature rely on the same physical model: only aerodynamic forces and moments proportional to the square of the propeller angular velocities are explicitly taken into account. Other aerodynamic effects are omitted and considered as small non-modeled disturbances to be rejected by the control law.

The reason put forward is that these effects are proportional to the square of the quadrotor linear velocity, hence very small near hover- ing. Few authors explicitly consider other aerodynamic effects : [12]

notes the importance of flapping stability derivatives; [13] and [14]

consider aerodynamic effects without physical motivation linear with respect to the quadrotor linear and angular velocities, but propose very small numerical values; [5] judges them to be negligible at low velocities, and focuses on nonlinear aspects at moderate velocities;

[15] physically motivates the presence of effects that are nearly linear with respect to the quadrotor linear and angular velocities, but pro- vides no experimental data and is concerned only with the open-loop system.

Figure 1 - Our home-built quadrotor : the “Quadricopter”

On the other hand, the accelerometer measurement vector a can be used in two different ways (gyros are used in both cases ; see page 4 for more details about inertial sensors):

1) as an input, directly in the equation V g a  = + if extra sensors pro- viding position or velocity information are available, using a sensor fu- sion algorithm that estimates the velocity and the pitch and roll angles

2) as an output, through the approximation a≈ −g. Accordingly, the pitch and roll angles are estimated by a sensor fusion algorithm.

Commercial “attitude sensors”, such as the 3DM-GX¹ or the MTi², run exactly on this principle.

In both cases, the sensor fusion algorithm can be an Extended Kalman Filter (EKF), a complementary filter, linear or nonlinear, or a nonlinear observer; see for example [16], [17] for an account of the two cases.

Recall that MEMS inertial sensors are not accurate enough for "true"

Schuler-based inertial navigation, see for example [18, Chap. 5] for details.

Now, a puzzling issue arises: the "conventional" physical model im- plies that the longitudinal and lateral (in body axes) accelerometers should always measure zero, which clearly contradicts 2) ; as for 1), even if no particular form of the accelerometers measurements is assumed, one may wonder about the interest of using measurements known to be zero (in addition corrupted by noise and biases). Never- theless, many successful quadrotor flights have been reported, with control laws based on 1) or 2), or even both, and there is no question that using accelerometers is beneficial.

This paper, which largely draws on [19], proposes a "revisited"

model containing extra aerodynamic terms proportional to the propel- ler angular velocity times the quadrotor linear or angular velocity. In particular, the so-called rotor drag, though rather small, appears at first order and is essential to correctly account for the accelerometer measurements.

The paper is structured as follows: the revisited model is derived in next section ; its main features are experimentally validated then ; finally, its implications for control schemes are discussed.

A revisited quadrotor model

Model of a single propeller “near” hovering

We first consider a single propeller rotating with angular velocity ε ω_{i i} around its axis k ; _b ω_i is positive, with ε₁=1 (resp. -1) for coun- terclockwise (resp. clockwise) rotation. Due to the motion of the quadrotor, the geometric center A₁ of the propeller moves with lin- ear velocity V_Ai

, while the rotor plane (by definition perpendicular to kb

) undergoes angular velocity ^Ω_^; the total angular velocity of the propeller is thusΩ+ε ω_{1 i b}\k

. A lengthy derivation in the spirit of for example [20, in particular Chap. 5] shows that the aerodynamic ef- forts on the propeller resolve into the force F_i

and moment M_i at A_i,

2

i i b i 1 Ai 2 b

F = a k − ω  −ω λ( V +^⊥ λΩ×k ) −ε ω λ_{i i}( V ×k +_{3 A}_i _b λ₄Ω^⊥)

(1)

2

i i

i i b i i 1 A 2 b

M = b − ε ω k −ε ω µS ( V +^⊥ µΩ× k ) −ω µ₁( V_{3 A}_i×k +_b µ₄Ω^⊥)

(2) where a,b, the λ_i′s and µ s are positive constants ; the projection of _i^, a vector U

on the rotor plane is denoted by

( ) ( )^·

b b b b

U^⊥ =k × U k× = −U U k k   

Moreoverλ₂andµ₂are very small (they would be exactly zero if the blade axis were orthogonal to k_b

). Notice that all of the force and moment terms orthogonal to k_b

arise from the velocity imbalance of the blade on a complete turn (because of the translational motion, the blade moves faster with respect to free air when it is advancing than when it is retreating).

The above relations rely on classical blade element theory, with two extra assumptions:

• the propeller is considered to be perfectly rigid, which is approxi- mately true for most quadrotor propellers. The flapping due to the slight flexibility of a real propeller has only a marginal effect ;

• the components of the linear velocity

Ai

V

are considered small with respect to the propeller tip speed ; similarly the components of the angular velocity Ω are considered to be small with respect toω_i. This is valid "near" hovering, i.e., for "small"

Ai

V

and Ω : typically, the tip speed is of about 50 .m s⁻¹, so that 10 .m s⁻¹can still be seen as a "small" velocity.

The velocities in the previous equations are of course velocities with respect to the air stream, not with respect to the ground. They coin- cide when there is no wind, which we assume in the sequel.

The term ω_iλ₁V_A^⊥_i

in (1) is often called H-force or rotor drag in the helicopter literature. Also notice that the simplified expressions (1)- (2), though directly based on textbook helicopter aerodynamics, do not seem to appear in the literature under this compact form, very handy for control purposes. The reason for this is probably that heli- copter literature is primarily concerned with articulated and/or rather flexible propellers, operating moreover at much higher ratios of linear velocity to propeller tip speed.

Figure 2 - Sketch of the complete quadrotor Complete quadrotor model

The quadrotor consists of a rigid frame with four propellers, (directly) driven by electric motors, see figure 2. The structure is symmetrically

1 www.microstrain.com

2 www.xsens.com

F4

F3

F1

A2

A4

A3

A1

F2

k0 j⁰ i0

kb

C

ib

 ib











arranged, with one pair of facing propellers rotating clockwise and the other pair rotating counterclockwise. The four propellers have the same axis k_b

.

3 1 3 1

b A A

i = A A



  

 , ^{4 2}

4 2

b A A

j A

= A



 

  and k_b

thus form a direct coordinate frame.

Let A be the geometric center of the A's₁ and 1 3 1 1 2 4;

2 2

l= A A = A A clearly, ⁴

1 i

i₌ AA =0.

∑ ^

The whole systemB, with mass m and center of massC, thus involves five rigid bodies : the frame/stator assembly B₀ and the four propeller/motor assembliesB_i. Clearly CA h= k_b

for some (signed) lengthh; notice that for most quadrotor designs h is very small. Resolved in the (i ,j ,k _{b b b}) frame, the velocity of C is written as V =ui +uj ivk_C  _{b b}+ _b and the angular velocity of B₀ is written as

b b b

pi qj rk Ω = +  + 

We assume that the only efforts acting on B are the weight and the aerodynamic efforts created by the propellers, as described in the previous section. In particular, we neglect the drag created by the frame, which is quadratic with respect to the velocity, hence small at low velocities with respect to the rotor drag. Newton’s laws for the entire system are thus written as

4

C 1 i

i

m V = m g + F

∑=

   (3)

4

C 1 i i i

i

= CA× F + M σ

∑= ^{  }

^B (4)

where σ_CB =∫ CM ×C^•Md Mµ( )

B is the kinetic momentum of B.

For each B_i, we can further write

i

i b i b i i

A ·k =M ·k +

σ^B    ε Γ (5)

where _iⁱ ( )

i i

A A M A M d M i µ σ^_B =∫  × ^•

B is the kinetic momentum of B_iand Γi is the (positive) electromagnetic torque of the motor. For sim- plicity, we have considered A_i as the center of mass of B_i(in fact the two points are slightly apart). We also consider the Γ_i′s as the control inputs (it is nevertheless easy to include the behavior of the electric motors, both for modeling and control).

We now evaluate the right-hand sides of (3)-(4). Since

Ai C i C b i

V =V +CA+AA   =V +h ×k + ×AA Ω  Ω 

we have

4

) 4 3 Ai b

3 C b i b

3 C b 4 3 i

V ×k

= ( V +h k AA k

= V k + ' +r AA

λ λ Ω

λ Ω Ω λ Ω

λ λ Ω λ

⊥

⊥

⊥

+

× + × × +

×



 

  

   



  

2

) ( ) 2

1 Ai b

1 C b i b

1 C 2 b 1 i b

V + k

= (V +(h k AA k

= V + ' +k - r AA k

λ λ Ω

λ Ω Ω λ Ω

λ λ Ω λ

⊥

⊥ ⊥ ⊥

⊥

×

× + × + ×

×

  

  

   



 

 

where we have used the fact that AA_i

is collinear to either i_b or j_b

, and setλ'₂=λ₁+hλ₁ and λ'₄=λ₄+hλ₃. Therefore,

4 4 4

2 1 2

1 1 1

4

3 4

1

4 4

1 3

1 1

4 4

2 1 2

1 1

( ' )

( ' )

( '

i i b i C b

i i i

i i C b

i

i i b i i

i i

i b i C b

i i

F a k V k

V k

r AA k r AA

a k V k

ω ω λ λ Ω

ε ω λ λ Ω

λ ω λ ε

ω ω λ λ Ω

⊥

= = =

⊥

=

= =

⊥

= =

   

= −   −  + ×

   

 

−  × +

 

   

+  × −  

   

   

≈ −     + ×

   

∑ ∑ ∑

∑

∑ ∑

∑ ∑

 

  

  

  

   

)

In the last line, we have neglected small terms according to the sec- ond extra assumption of the single properller model. Indeed, in hover- ing V_C and Ω

, hence V_Ai

are zero ; from (1)–(4) this implies that

2 2 2 2

1 2 3 4

( )

aω +ω +ω +ω =mg

and ω₁²−ω₂²+ω₃²−ω₄²=ω₁²−ω₃²=ω₂²−ω₄²=0, and eventually

i mg4

ω = =ω a .

As a consequence _i⁴_{1 i i},1 _i⁴_{1 i}A _i

l A

ε ω ω

= =

∑ ∑ ^{and 4}1

1 _{i i i},AA_i l∑₌ ε ω ^{}

also vanish in hovering ; “near” hovering they are therefore small with respect to∑⁴_i=1ω_i^.

Similar computations yield

4

1

4 4

2 2

1 1

4 4

12 1 1 3 4

1 1

( ' " )

i

i i i i

i

i i b i i b

i i

b C b

i i

CA ×F +AA ×F M

a AA ×k b k

r l k V ×k

ω ε ω

λ ω ω µ µ Ω

=

= =

⊥

= =

+

   

≈ −   −  

   

   

−   −  +

   

∑

∑ ∑

∑ ∑

    

  

   

whereµ'₃=µ₃− hλ₁andµ"₄= + h(µ₄ µ₁+λ' )₂ .

Notice that the contributions of λ λ₃, ₄ in the forces (1) and of µ µ₁, ₂ in the moments (2) (nearly) cancel out in the right-hand sides of (3)- (4), due to the fact there are two clockwise and two counterclock- wise-rotating propellers.

We then evaluate the left-hand sides of (3)–(5). The approach is fairly standard.

1

4 1

4 1

(

O

O

i

C

i

i i i b i

i

i i

= CM× d (M)

= CM× d (M)

+ CM×( + ) d M)

= CM×( ×CM ) d (M)

+ CM×( ×CA +( + k )×A M CM

CM

CA A M

) dµ(M)

= CM×( ×

σ µ

µ

µ

Ω µ

Ω Ω ε ω

Ω

•

•

• •

=

+

∫

∫

∑ ∫

∫

∑ ∫

∫









  





  

 



 





 



B B

B

B

B

B

B

1

4 1

4 1

4 1

( .

(

i i

i i i b i

i

C i i i A b

b b r i i i b

CM )d (M)

+ A M×( k ×A M ) d M)

= ( .k )

=Ipi Iqj Jr J )k

µ

ε ω µ

Ω ε ω

ε ω

=

=

=

+

+ + +

∑ ∫

∑

∑



  





  

 

B

B B

where I; J; J_r are strictly positive constants. In the last equation, in the computation of the inertia tensors ⁱ

Ai

,

^B ^B we have replaced

In (6)–(7) we have assumed λ'₂= + hλ₂ λ ≈₁ 0, which is sensible since λ₂ and h are nearly 0 (notice that λ ='₂ 0 can always be enforced by slightly shifting the center of mass). Finally, the angles and angular velocities are linked by

cos

= p+(q sin +r cos )tan

=qcos r sin q sin +r cos

=

φ φ φ φ

θ φ φ

φ φ

ψ θ

 −



Equations (6)–(15) form the complete 13-dimensional nonlinear model of the quadrotor.

A further simplification is to replace ∑⁴_i=1ω_i^{by 4ω} in (6) – (12) since ω_i remains close to w in normal flight and moreover use the fact that the propeller moment of inertia J_r is very small with respect to J I− ; this yields

1

1

4 2

1

2 2

4 2 3 4

2 2

4 2 3 4

2 4

1 1

2 2

1 1

sin 4 sin cos 4 cos cos

( ) ( )4 ( ' " )

( ) ( )4 ( ' " )

4

4 , 1,

i i

i i i

r i i i

u qw rv g u

m

v ru pw g v

a m w pv qu g

m

Ip J I qr a v p

Iq J I qr a u p

Jr l r

J l Jrr b i

J

θ ωλ φ θ ωλ

φ θ ω

ω ω ω µ µ

ω ω ω µ µ

ω λ ε Γ

ω ε ωλ Γ ω

=

=

+ − = − −

+ − = − −

+ − = −

+ − = − +

+ − = − +

= − −

− = − =

∑

∑













 2,3,4

Equations (16)–(22) can be used instead of (6)–(12) with no notice- able loss of accuracy.

Model of the inertial sensors

The quadrotor is equipped with strapdown triaxial gyroscope and ac- celerometer. Without restriction, we assume that the sensing axes coincide withi j k _b, , ._b _b

The gyroscope measures the angular velocity Ω

, projected on its sensing axes, i.e., (g ,g ,g )=(p, q, r),_{x y z} the accelerometer measures the specific acceleration a = V _P−g of the point P where it is located, projected on its sensing axes; see for example [18, Chap. 4] for details on inertial sensors. Hence, by (3) if the accelerometer is located at the center of mass C, which is the case for most quadrotors, it measures

4 1

C 1 i i

a = V g= F

m ⁼

− ∑

 

  

by (3), the accelerometer thus measures

1 1 2 3 4

1 1 2 3 4

2 2 2 2

1 2 3 4

. ( )

. ( )

. ( )

x b

y b

z b

a a i u

m

a a j v

ma a a k

m

λ ω ω ω ω λ ω ω ω ω

ω ω ω ω

= = − + + +

= = − + + +

= = − + + +





  the actual propellers by disks with the same masses and radii, and

taken advantage of the various symmetries ; this "averaging" approxi- mation is justified by the fact that the propeller angles vary much faster than all of the other kinematic variables (besides, this approxi- mation is already heavily used in the blade element theory used to derive (1)-(2)). Using the same approximation,

( )

( ) ) ( )

.( )

( )

i

i i

i i i

i i

A

i i i b i

i i b A

r b r b r i i b

= A M A M dµ M

= A M k A M dµ M

= k

=I pi I qj J r k σ

Ω ε ω Ω ε ω

ε ω

×

× + ×

+

+ + +

∫

∫

 



   





  

B B

B

B

where I_r is a strictly positive constant. Eventually,

4 1 4

1 4

1

( )

.

. ( )

.

i i

C b C b

C b

r i i i

C b

C b r i i i

C b r i i i

C

V ·i u qw ru V . j = v ru pw w pu qu V · k

Ip J I qr J q i

j = Iq J I pr J p k Jr J

σ ε ω

σ ε ω

σ ε ω

σ

=

=

=

 

+ −

 

 

 

   + − 

   + − 

   

 

 

 + − + 

   

   + − + 

   

   

   + 

   

∑

∑

∑

 

  

 

 

  

 

 

   



B B B

B.k = J r_{b r}( +ε ω_{i i}) i=1,2,3,4





To describe the orientation of the quadrotor, we use the classical φ θ Ψ, , Euler angles (quaternions could of course be used). The di- rection cosine matrix R_{φ θ Ψ, ,} to convert from Earth coordinates to aircraft coordinates is then

C C C S S

S S C C S S S S C C S C C S C S S C S S S S C C

θ ψ θ ψ θ

φ θ ψ φ ψ φ θ ψ φ ψ φ θ φ θ ψ φ ψ φ θ ψ φ ψ φ θ

 − 

 − + 

 

 + − 

 

so that g g i= −( sinθ+ j sin cosφ θ+k cos cos )φ θ . Collecting the previous findings (3)–(5), we eventually have

1 4 1 1 4

1

4 2

1 4

1

2 2 ' '' 4

1 3 3 4 1

4 1

2 2 '

3 i i

i i

i i

r i i i

i i

r i i i

1 3

u+qw rv = g sin u m v+ru pw = g sin cos v

a m w+pv qu = g cos cos

m Ip+(J I)qr+J q

= a( - )+( v+ p) Iq (J I)pr J p

=a( )+( u

θ λ ω

φ θ λ ω

φ θ ω

ε ω

ω ω µ µ ω

ε ω

ω ω µ

=

=

=

=

=

=

− − −

− −

− − −

−

− − −

− −

∑

∑

∑

∑

∑

∑











'' 4

4 1

4 4

1 1 1

2

i i

r 2 i i i i i

r i i i i

q) (J 4J )r = l

J ( r+ )= b i=1,2,3,4

µ ω

λ ω ε Γ

ε ω Γ ω

=

= =

− − −

−

∑

∑ ∑





(6)

(7) (8)

(9)

(10) (11) (12)

(13) (14) (15)

(16) (17) (18) (19) (20) (21) (22)

(23) (24) (25)